Transcript Factoring out the GCF

To factor means to write a number or
expression as a product of primes.
In other words, to write a number or
expression as things being multiplied
together.
The things being multiplied together are called
factors.

Here is a simple example of factoring:
12  2  2  3
The factors are 2, 2 and 3 which are all
prime numbers. (They are only divisible by
themselves and 1.)
Factoring is the opposite of simplifying.
x 2  6 x  18
To go from x 2  5 x  14
to
(x+7)(x-2)
you simplify.
you factor.
To go from (x+3)(x-6)
to
(3x  2)( x  5)
Factoring
Simplifying
3 x  13 x  10
2
No parentheses and no like
x  16 Simplified.
terms.
( x  4)( x  4) Factored. A product of primes.
2
A product of primes.
2 x y (3x  6 y ) Factored.
There are 4 factors.
2
2, x 2 , y and (3x  6 y)
6 x y  12 x y
3
2
2 Simplified. No parentheses and
no like terms.
The first thing you always do when factoring is
look for a greatest common factor.
GCF
GCF: the biggest number or expression that all
the other numbers or expressions can be
divided by.
What is the GCF of 27 and 18?
The biggest number they are both divisible by is
9 so the GCF is 9.
What is the GCF of 16x2 and 12x?
The biggest number that goes into 12 and 16
is 4.
The biggest thing x2 and x are divisible
by is x.
The GCF is 4x.
** If all the terms contain the same variable, the GCF will contain the
lowest power of that variable.
Factoring out or pulling out the GCF is using
the distributive property backwards.
3 x( x  6)  3x  18 x
2
Distribute 3x
3 x  18 x  3 x( x  6)
2
factor out 3x
Factor
5 x  10 x  25 x
4
3
2
1. Find the GCF
GCF = 5x2
2. Pull out the GCF
5x2(____ - ____ + ____)
3. Divide each term by the GCF
to fill in the parentheses.
5 x 4 10 x3 25 x 2 2
 2  2  x  2x  5
2
5x 5x
5x
5x  10 x  25x  5x ( x  2 x  5)
4
3
2
2
2
Distribute to check your answer.
Factor 16a
b  14a b  4a b
2 3
5 2
8
1. Find the GCF
GCF = 2a2b
2. Pull out the GCF
2a2b(____ - ____ + ____)
3. Divide each term by the GCF
to fill in the parentheses.
16a 2b3 14a5b2 4a8b
2
3
6



8
b

7
a
b

2
a
2 a 2 b 2 a 2 b 2a 2 b
16a b  14a b  4a b  2a b(8b  7a b  2a )
2 3
5 2
8
2
2
3
6
Factor
2 x ( x  5)  3( x  5)
2
1. Find the GCF
GCF = (x + 5)
2. Pull out the GCF
(x + 5)(_____ - _____)
3. Divide each term by the GCF
to fill in the parentheses.
2 x2 ( x  5) 3( x  5)


( x  5)
( x  5)
2x  3
2 x ( x  5)  3( x  5)  ( x  5)(2 x  3)
2
2
2
Factor
13x  10 y
3
2
1. Find the GCF
HMMMM?
These two terms do not have a common factor other than 1!
If an expression can’t be factored it is prime.
You try: Factor
m  7 m  4m
5
3
4
1. Find the GCF
2. Pull out the GCF
3. Divide each term by the GCF
to fill in the parentheses.
m  7m  4m  m (m  7  4m)
5
3
4
3
2
Better written as-
m3 (m 2  4m  7)
You try: Factor
3x( x  7)  2( x  7)
1. Find the GCF
2. Pull out the GCF
3. Divide each term by the GCF
to fill in the parentheses.
3x( x  7)  2( x  7)  ( x  7)(3x  2)
Factoring Binomials
(2 Terms)
Remember:
The first thing you always do when factoring is
pull out a GCF if possible.
If there is a GCF and you have factored it out you
then look to see if there is any other factoring
that can be done.
When you have to factor an expression with
two terms it could be a difference of squares
or a sum or difference of cubes.
A difference of squares involves two terms
that are both perfect squares and
subtraction.
A sum or difference of cubes involves two
terms that are both perfect cubes. The
operation between them can be addition or
subtraction.
Difference of Squares
perfect square – perfect square
Perfect Squares:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144,
169…
If a variable has an even exponent then it is a
perfect square.
2
4
16
x , x , x ...
Difference of Squares
A difference of squares factors as binomial conjugates. There
will be two binomial factors containing the same terms but one
will be addition and one will be subtraction.
You factor a difference of squares using the
following rule:
a2
a  b  (a  b)(a  b)
2
2
b2
addition
subtraction
Factor:
4 x  64
2
4x 2
(2 x  8)(2 x  8)
64
Check your answer by FOILing!
Sum and Difference of Cubes
Sum of cubes and difference of cubes are
both binomials.
Both terms are perfect cubes.
perfect cube + perfect cube
or
perfect cube – perfect cube
Perfect cubes: 1, 8, 27, 64, 125… or any
variable with an exponent divisible by 3.
Sum and Difference of Cubes
A sum or difference of cubes
will have two factors. One is a
binomial the other is a
trinomial.
Factor a sum or difference of cubes using the
following rule:
a  b  a  ba  ab  b
3
3
a b
3
3

 a  ba  ab  b 
Notice the terms in the binomial
factor are the cubed root of the
terms in the original problem and
the sign is the same.
2
2
2
2
You can remember this by
remembering CSC (cubed root,
same sign, cubed root)
a  b  a  ba  ab  b
3
3
a b
3
3

 a  ba  ab  b 
Notice the first and last terms in
the trinomial factor are the square
of the terms in the binomial
factor.
2
2
2
2
Notice that the middle term in
the trinomial factor is the
product of the two terms in the
binomial factor.
a  b  a  ba  ab  b
3
3
a b
3
3

 a  ba  ab  b 
2
2
2
2
Notice that the first sign
in the trinomial factor is
the opposite of the sign
in the binomial factor.
Notice that
the last
operation
is always
addition.
a  b  a  ba  ab  b
3
3
a b
3
3

 a  ba  ab  b 
2
2
2
2
So, you can remember what goes in the
binomial factor by remembering CSC
(
Cubed root, Same sign, Cubed root)
a  b  a  ba  ab  b
3
3
a b
3
3

 a  ba  ab  b 
2
2
2
2
To remember what goes in the trinomial factor just
remember SOPAS
Square, Opposite sign, Product, Add, Square